Analysis of Knuth's Sampling Algorithm D and D'
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In this research paper, we address the Distinct Elements estimation problem in the context of streaming algorithms. The problem involves estimating the number of distinct elements in a given data stream 𝒜 = (a_1, a_2,…, a_m), where a_i ∈{1, 2, …, n}. Over the past four decades, the Distinct Elements problem has received considerable attention, theoretically and empirically, leading to the development of space-optimal algorithms. A recent sampling-based algorithm proposed by Chakraborty et al.[11] has garnered significant interest and has even attracted the attention of renowned computer scientist Donald E. Knuth, who wrote an article on the same topic [6] and called the algorithm CVM. In this paper, we thoroughly examine the algorithms (referred to as CVM1, CVM2 in [11] and DonD, DonD' in [6]. We first unify all these algorithms and call them cutoff-based algorithms. Then we provide an approximation and biasedness analysis of these algorithms.
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